Biblical numbers, mathematics and attributed patriarchal ages

The Hebrew Bible is filled with numbers. There are different kinds of numbers -- cardinals and ordinals, integers and fractions, even primes. And they are everywhere in the Torah text.

There are numbers for days and numbers for life spans.

There are numbers for populations and numbers for the duration of events.

There are numbers for the measurement of quantities and numbers for the sizes of objects and areas.

There are numbers for the duration of events.

And there are numbers for a host of seemingly mundane things, such as the number of visitors and the number of palm trees.

Some Biblical numbers are curiously round. For instance, Noah reportedly was 500 years old when his son Shem was born, and 600 when the great Flood occurred. Shem was, then, 100 years old at the time of the Flood, and he died 500 years later at age 600.

And Isaac married Rebekah at age 40, became a father of twins at age 60 and was 100 when Esau was married. He died at age 180.

Some Biblical numbers are identical and strangely coincidental. The rain that created the Flood for Noah lasted 40 days and nights. That is the same number of days and nights that Moses reportedly spent on Mount Sinai on each of two occasions to receive the tablets engraved with the teachings and the commandments.

Other numbers appear hyperbolic and incredible. The Bible states that Jacob’s descendants consisted of 70 individuals when the family entered Egypt. After centuries of involuntary servitude, the number of adult males that left Egypt with Moses is asserted to be about 600,000. Including wives, concubines and children, the total number of those leaving must have been in the millions. Understood literally, the number seems absurdly large.

When we encounter numbers in the Bible, what are we to do with them? Are all or some of the numbers to be taken literally or is one or more of them to be understood symbolically? And how can we tell which numbers fall into which category? Do we have a collection of essentially random numbers? Or, are there patterns that provide information, suggest meaning, or, maybe, reveal secrets?

Of all the number puzzles in the Bible, perhaps none is more intriguing than the longevity of the generations from Adam though Moses. Using a variety of approaches, scholars and others have long considered the numbers found in the Bible. They have speculated fancifully in an effort to make sense of some of them. With perhaps rare exceptions, though, these efforts have not been particularly satisfying, leaving the original problem, as the mathematician Lewis Carroll had Alice say in a different context, “curiouser and curiouser!” (See Carroll, Alice’s Adventures in Wonderland (Ch. 2.))

The data

Let’s look at the data first. According to the Torah text (meaning here the Masoretic text), there were 10 generations from Adam to Noah, inclusive. There were another 10 generations after Noah and to Abraham, inclusive. Coincidence?

The life spans recorded in the Bible for the first group are as follows: Adam (930), Seth (912), Enos (905), Cainan (910), Mahlaleel (895), Jared (962), Enoch (365), Methuselah (969), Lamech (777) and Noah (950). The life spans recorded for the second group, exclusive of Terah’s son Abraham, are as follows: Shem (600), Arphaxad (438), Salah (433), Eber (468), Peleg (239), Reu (239), Serug (230), Nahor (148) and Terah (205).

Looking at the three primary patriarchs separately, Abraham, Isaac and Jacob lived 175, 180 and 147 years respectively. Parenthetically, the only life span we have for a matriarch is that for Sarah, who died at age 127.

Finally, while at one point the Bible states that Jacob’s descendants were in Egypt for 430 years (see Ex. 12:40-41), the book of Exodus records just four generations from Jacob to Moses. Jacob’s son Levi lived 137 years, Levi’s son Kohath lived 133 years and Kohath’s son Amram lived 137 years. (Ex. 6:16, 6:18, 6:20.) Moses, who was Amram’s son and, therefore, Jacob’s great-great grandson, lived 120 years. (Deut. 34:7; see also, The Torah: A ModernCommentary (URJ Press 2005), at 383 n. 20.)

What, if anything, do these 26 numbers tell us?

Attempts to explain Biblical ages

Even a quick review suggests that the life spans of the first group of Biblical humanity was reasonably consistent within a relatively narrow range of 895-969 years, with the notable exceptions of the seventh generation descendant Enoch and Noah’s father, Lamech. As a general matter, the life spans of the second group, and through Moses, then continued on a downward slope. The first three individuals after Shem lived between 433 and 468 Biblical years, and the next three lived in a reduced range of 230 to 239 Biblical years. After Terah, no Biblical personality is indicated to have lived in excess of 200 years.

Rabbinic sages of the past accepted the reported ages as accurate. To explain the longevity of Adam, one noted that Adam was made in God’s image and therefore physically perfect. Adam’s immediate descendants were similarly vigorous. To explain the decline after the Flood, one sage suggested that the Earth’s atmosphere deteriorated. Another opined that the people faced a harsher climate after the post-Babel dispersion. (SeeThe Chumash (Stone Ed. Mesorah 1993) at 25, 51 n. 19.) There is, unfortunately, no actual evidence to support such speculation, no way to test any of these propositions.

A more modern and purely mathematical analysis of the numbers by Charles A. Glatt, Jr. concludes that the longevity of the individuals from Noah to Joseph (and then to Moses) declined in a manner consistent with base e exponential decay (where e = 2.718 . . .). (See Glatt, “Patriarchal Life Span Exponential Decay by Base e,” at www.ldolphin.org/lifespans.pdf.) Base e is a rate of growth (or decay) that appears to exist commonly in nature in continuing processes. If Glatt’s analysis is solid, then it would not only provide a naturalistic explanation for the reduction of the life spans studied, but might also impart some credence to the ages themselves.

Glatt used the ages of 14 individuals from Noah through Joseph, inclusive, as the raw data for his study. He claims that ten of the fourteen data points fall with one estimate of the standard deviation and the other four come within two standard deviations. (Id. at 9.) While the author concedes that his equation does not work if extended to today, he does argue that it “does fit the data for the 1500 years from the flood to Moses . . . .” He reaches this conclusion by looking at the calculated lifespan for the time in which Moses lived (said by Glatt to be 2,580 years after Adam) which yields 68 years, and he implies a consistency with the 70 year lifespan asserted in Psalm 90, the Psalm of Moses. (Id.)

Initially, there is a general concern with Glatt and his analysis. His bona fides are not evident. He provides no background information so that his qualifications can be checked and no contact information so that he may be called upon to respond to inquiries. His paper is not even dated. Still, treating the analysis as it appears, is Glatt on to something, or he is merely trying to squeeze a result out of the data?

There are both general and specific questions to be asked here. As a general matter, why is decay analysis an appropriate form of investigation of population ages? Isn’t this sort of exponential analysis more suited to radioactive decay, interest calculations or the growth of crystals or populations? And if decay analysis is appropriate generally, are fourteen data points really enough information on which to base a serious study?

There are specific issues, as well. One major error appears to be in the time frame used by Glatt. He has Terah born in 1879 AC (After Creation, sometimes Anno Mundi) and Terah’s son Abraham born in 2009 AC, when Terah would have been 130 years old. (Id. at 8.) According to Gen. 11:26, however, Terah was 70 years old at Abraham’s birth, meaning that Abraham was born in 1948 AC. All subsequent dates are erroneous as well. (For a traditional chronology and timeline, seeThe Chumash, above, at 53.)

Two other problems relate to the predicted and the reported duration of Moses’ life. First, and most simply, according to the Torah text, as noted above, Moses lived to be 120 years old. Glatt makes no effort in his study to address that number or relate it to his study. Second, Glatt does not explain how he reached either the 2,580 year number for “after Adam to Moses” or the 1,500 year number “from the flood to Moses.” Moreover, if Adam died in 930 AC, then according to Glatt Moses would have been alive in 3,510 AC, but if Moses were alive 1,500 years after the Flood, which occurred in 1656 AC, then he was alive in 3,156 AC. Not only are those results inconsistent, if the current count of years after Biblical creation comes to 5774, then both of Glatt’s timeframes place Moses much closer to the common era than seems justified.

Glatt’s real focus, though, is from Noah to Joseph. Within that range, though, Glatt does not express any opinion as to why the numbers seem to fall in line with his equation. Is he simply noting or claiming implicitly claiming that a natural phenomenon was at work. Or is he implicitly contending that supernatural activity was involved? Regardless of the answers to these questions, if one accepts the notion that base e decay was at work, how does Glatt explain why there are any deviations from the path predicted by the decay equation?

Finally, and despite the obvious decline in life spans, the raw numbers in group 2, and through Moses, still seem high. After all, the ages at the respective deaths of the fourteen identified individuals, Glatt’s data points, never fall below 110. Other than an unsupported statement (at 3), that “a global longevity of around 100 years is historically recorded in Egypt around the time of Joseph,” Glatt does not provide any information on life spans generally for populations 3,000-5,000 years ago.

Of course, accepting the accuracy of the given numbers, whether done so by rabbis centuries ago or mathematicians today, is problematic at another level. It avoids, even precludes, the possibility that the numbers we read or hear mean something other than what they appear to mean.

In connection with the antediluvian life spans recorded in the Bible, the editors of The Torah: A Modern Commentary, published by the Union for Reform Judaism, refer to the Metonic Cycle, a relationship of the solar and lunar cycles bearing the name of an Athenian astronomer Meton in the 5th century BCE, but, apparently, recognized long before him. The Meton Cycle is premised on the equivalence in time of 235 lunar cycles and 19 solar years. The editors then note that certain life spans of the ancient aged, like Methuselah’s 969 years and Noah’s 950, are divisible by 19. Under this approach, Methuselah lived 51 Meton Cycles and Noah lived 50. (See The Torah: A Modern Commentary, above, at 42.)

But what good is this information? If it is meant to suggest a formula by which we can devise the true age of these Biblical individuals, then it does bring the numbers down to an area where we are more comfortable, but applying the formula creates other problems. Noah was 500 Biblical years old, or 26.3 Metonic Cycles, when he fathered Shem, but Methuselah was 187 Biblical years old when his son Lamech was born. Was Methuselah just under 10 in Metonic Cycles? What about Mahalaleel who was 65 biblical years old when his son Jared was born? That comes to just over 3 Metonic Cycles.

Umberto (Moshe David) Cassuto (1883-1951 CE) was the director of the Rabbinic Seminary in Florence Italy and, starting in 1939, chair of Bible studies at Hebrew University in Jerusalem. Cassuto has argued generally that it is not possible to comprehend Torah without reference to Israel’s neighbors. More specifically, he has contended that the ancient chronologies, and the use of numbers generally in the Bible, cannot be understood without reference to the sexagesimal system which prevailed in ancient Mesopotamia, often supplemented with the use of the number 7 or some multiple of it. (See generally, Cassuto, From Adam to Noah (Magnes, Eng. Ed., 1961), at 1,7, 192-93, 258-62; From Noah to Abraham (Magnes, Eng. Ed., 1964), at 255, 257-62.)

The sexagesimal system placed great emphasis on the number 6 and extensions of that number, including 60, 600, 6,000, 60,000 and 600,000. Fractions and multiplications of those numbers were also important. So, if 60 was a value, then 2 times 60, or 120, was even more special.

The numbers 5 and 7 were also important. According to Cassuto, 5 years was the equivalent of 60 months. The number 5 factored, i.e., the product of 1 times 2 times 3 times 4 times 5 turns out to be 120. The number 7 was considered an ideal or perfect number, one that was a sign of totality or completeness. (See Cassuto, above, From Adam to Noah, at 12-17, 191-92, 243, 256, 258-62, 276; see also, Gen. 2:1-3.) The number 7 appears by itself hundreds of times in a wide variety of contexts, and as an additional element or factor, such as 17 or 77, in scores more.

For Cassuto, then, the years listed in Genesis for the first 10 generations, with two special and explainable exceptions, can be divided into two groups, one where the ages end in 5 or a multiple of 5 and another where seven has been added to such a number. As he notes, the reported life spans of six of the first ten individuals identified from Adam through Noah end either in 5 or a multiple of 5 which yields an end number of 0. Two more ages fit the pattern if 7 is added. Methuselah’s age of 969 requires the addition of 2 times 7 to conform, perhaps further underscoring the exceptional importance of his age.

The two individuals whose ages do not fit the main pattern are, to Cassuto, the proverbial exceptions that prove the rule. Methuselah’s father, Enoch, lived only 365 years. The age 365 can be viewed as special in at least three different ways. It may refer to the number of days in a solar year. It may be seen special under the sexagesimal system, being the total of 6 times 60 plus 5. It is also a schematic number, the sum of 102 plus 112 plus 122. We do not know what the number denotes in this particular instance, but it does not seem merely coincidental that this 7th generation person was singled out for special treatment. Lamech’s age of 777, as described in Torah at Gen 5:31, is seven and seventy years and seven hundred years, an obvious emphasis on the ideal number 7, and, again, more likely symbolic than factual.

The ages of the second group of individuals, from Shem through Abraham do not fall so neatly into Cassuto’s system. Shem at 600 does and so does Terah at 205, but the others do not. Perhaps, as with Sherlock Holmes’s encounter with the hound of the Baskervilles, the clue is in the absence, rather than the presence, of something expected. Here, perhaps, the failure of the numbers to end in a 5 or a 0 signals that the member of this group are not as special or deserving as the original group of 10.

The primary patriarchs

What happens when we reach the ages of the three primary patriarchs? As set forth above, the life spans recorded for Abraham, Isaac and Jacob are 175, 180 and 147, respectively. Each number falls neatly within Cassuto’s analysis. The numbers 175 and 180 end in 5 and 0, respectively, and 147 reflects an addition of 7 to the product of 2 times 70.

Not incidentally, the approach works for the matriarch Sarah, as well. Torah tells us that Sarah died at age 127, which is 2 times 60 (a good sexagesimal number) plus 7. If 120 was an honorific number, then 127 was even more so.

The ages of the three primary patriarchs are, however, also part of a sequence, and a unique one at that. As illustrated by Nahum Sarna, each number in the sequence is the product of a number squared multiplied by another number. And the numbers involved follow in two clear descending and ascending orders. So, 175 = 7 times 52, 180 = 5 times 62, and 147 = 3 time 72. (See Sarna, Understanding Genesis (Schocken 1970), at 84.) That formula can be extended in two directions, resulting in 144 for 9 times 42 and 64 for 1 times 82. No patriarch or other biblical personage is assigned either number, however, underscoring how special were the ages attributed to the primary patriarchs.

In his marvelous review of Genesis, then University of Chicago Professor Leon Kass added, no pun intended, that the sum of the three equations is identical. That is, 5 plus 5 plus 7 is the same as 6 plus 6 plus 5 which equals 7 plus 7 plus 3. Each total is 17. As to what possible significance this could be, Kass says, however: “I have no idea.” (Kass, Beginning of Wisdom (Free Press 2003), at 629 n.18.)

And there lies the rub. We are so removed from the mathematical considerations that even when we think we can identify some of the signals, we cannot determine where they point. The number 17 is, of course, a prime number, one not divisible by any number other than itself and 1. Moreover, it is the 7th prime number after 1, 3, 5, 7, 11 and 13. It is also the sum of 7 and 10, the latter also considered a number of ordinal perfection.

The number 17 occurs in interesting and varied contexts. For instance, after he was born, Joseph lived with his father Jacob for 17 years until he was sold to the Ishmaelites or, alternatively, the Midianites. Later, Jacob lived with Joseph in Egypt for 17 years until he died. (See Gen. 37:2, 25-28, 47:28.) Further, the life span of Levi, another of Jacob’s son, was 137, which is the sum of 120 and 17.

What we do know, though, and what is important to know, is that the attributed ages of the primary patriarchs, are neither random nor mere numbers to be located as data points on a line. They are numbers, whether products or sums, which suggest an intent by the author or editor of the text to convey something important about the three primary patriarchs, their distinctiveness from all others and their relationship to each other.

A few final points need be made. First, this discussion has been limited to numbers that appear in the Masoretic text. There are some different numbers that appear in the Greek Septuagint and the Samaritan Pentateuch. (See Cassuto, From Adam to Noah, above, at 264-65; From Noah to Moses, above, at 258-59.)

Second, Cassuto was not a disinterested academic. In addition to desiring to show what Torah text meant to those who first heard it, he was interested in demonstrating that the text of Genesis was part of a unified whole, rather than a collection of writings of different authors as postulated by the documentary hypothesis of higher biblical criticism. (See, e.g., Cassuto, From Adam to Noah, above, at 94, 193.)

Nevertheless his insights cannot be denied. Sarna concludes that “the biblical chronologies of the patriarchal age are not intended to be accurate historical records in our sense of the term.” (Exploring Genesis, above, at 84.) Rather, they “fall within the scope of historiosophy, or philosophy of history, rather than historiography.” (Id.) In this view, they tell us less about actual time intervals than about the ideas of Biblical import. Numerical symmetry or harmony is not a matter of coincidence among random events, but a signal of importance, and a sign of presumed divine control and direction. (Id. at 85.)

Lastly, whoever wrote or redacted the Hebrew Bible was more than a drafter of national history, a recorder or developer of laws and mores and a masterful storyteller. He, she or they also had a real competency with mathematics. We do not yet understand the signals, or even whether they were theologically purposeful or narrative drivers or both, but they are clearly there, mean something and deserve serious and further consideration.

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